Chapter 7.3, Problem 52E

(a)

To determine

The volume of the column.

Answer to Problem 52E

The volume of the column is $s^{2}h$

Explanation of Solution

Given:

The side length of square is s which lies in a plane perpendicular to L. One vertex of the square lies on L.

Calculation:

The volume of column can be find out as:

  $A(x)={(\text{side length})}^2=s^2$

  $\begin{array}{l}V={\displaystyle \underset{a}{\overset{b}{\int }}A(x)dx}\\ V={\displaystyle \underset{0}{\overset{h}{\int }}{s}^{2}dx}\\ V=s^2{\left(x\right)}_0^h\\ V=s^{2}h\end{array}$

Conclusion:

The volume of the column is $s^{2}h$

(b)

To determine

The change in volume if square turns twice instead of one.

Answer to Problem 52E

There is no change in volume.

Explanation of Solution

Given:

The square turns one revolution about L.

Calculation:

The cross section area of the twisted column is same as the original column. Hence, both original and twisted have the same volume.

Now, if square turns twice then the new volume can be find out as:

  $s^2\frac{h}{2}+s^2\frac{h}{2}=s^{2}h$

This is the same volume as before, hence no change in volume if square takes two revolution instead of one.

Conclusion:

There is no change in volume.